\(\int e^{-3 \text {arctanh}(a x)} (c-\frac {c}{a x})^{7/2} \, dx\) [587]

Optimal result

Integrand size = 24, antiderivative size = 263 \[ \int e^{-3 \text {arctanh}(a x)} \left (c-\frac {c}{a x}\right )^{7/2} \, dx=-\frac {984 a^3 \left (c-\frac {c}{a x}\right )^{7/2} x^4}{5 (1-a x)^{7/2} \sqrt {1+a x}}-\frac {398 a^2 \left (c-\frac {c}{a x}\right )^{7/2} x^3}{15 (1-a x)^{3/2} \sqrt {1+a x}}+\frac {38 a \left (c-\frac {c}{a x}\right )^{7/2} x^2}{15 \sqrt {1-a x} \sqrt {1+a x}}-\frac {2 \left (c-\frac {c}{a x}\right )^{7/2} x \sqrt {1-a x}}{5 \sqrt {1+a x}}+\frac {427 a^3 \left (c-\frac {c}{a x}\right )^{7/2} x^4 \sqrt {1+a x}}{15 (1-a x)^{7/2}}+\frac {13 a^{5/2} \left (c-\frac {c}{a x}\right )^{7/2} x^{7/2} \text {arcsinh}\left (\sqrt {a} \sqrt {x}\right )}{(1-a x)^{7/2}} \] Output:

-984/5*a^3*(c-c/a/x)^(7/2)*x^4/(-a*x+1)^(7/2)/(a*x+1)^(1/2)-398/15*a^2*(c- 
c/a/x)^(7/2)*x^3/(-a*x+1)^(3/2)/(a*x+1)^(1/2)+38/15*a*(c-c/a/x)^(7/2)*x^2/ 
(-a*x+1)^(1/2)/(a*x+1)^(1/2)-2/5*(c-c/a/x)^(7/2)*x*(-a*x+1)^(1/2)/(a*x+1)^ 
(1/2)+427/15*a^3*(c-c/a/x)^(7/2)*x^4*(a*x+1)^(1/2)/(-a*x+1)^(7/2)+13*a^(5/ 
2)*(c-c/a/x)^(7/2)*x^(7/2)*arcsinh(a^(1/2)*x^(1/2))/(-a*x+1)^(7/2)
 

Mathematica [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.

Time = 4.33 (sec) , antiderivative size = 200, normalized size of antiderivative = 0.76 \[ \int e^{-3 \text {arctanh}(a x)} \left (c-\frac {c}{a x}\right )^{7/2} \, dx=-\frac {c^3 \sqrt {c-\frac {c}{a x}} x \left (-3 \sqrt {-a x (1+a x)} \left (-6921+19192 a x-21508 a^2 x^2-28706 a^3 x^3+6325 a^4 x^4-2470 a^5 x^5+520 a^6 x^6\right )+585 \left (-35+70 a x-86 a^3 x^3+19 a^4 x^4\right ) \arcsin \left (\sqrt {-a x}\right )+1040 a^4 x^4 (-1+a x)^3 \sqrt {1+a x} \sqrt {-a x (1+a x)} \operatorname {Hypergeometric2F1}\left (\frac {3}{2},\frac {9}{2},\frac {11}{2},-a x\right )\right )}{720 (-a x)^{7/2} \sqrt {1+a x} \sqrt {1-a^2 x^2}} \] Input:

Integrate[(c - c/(a*x))^(7/2)/E^(3*ArcTanh[a*x]),x]
 

Output:

-1/720*(c^3*Sqrt[c - c/(a*x)]*x*(-3*Sqrt[-(a*x*(1 + a*x))]*(-6921 + 19192* 
a*x - 21508*a^2*x^2 - 28706*a^3*x^3 + 6325*a^4*x^4 - 2470*a^5*x^5 + 520*a^ 
6*x^6) + 585*(-35 + 70*a*x - 86*a^3*x^3 + 19*a^4*x^4)*ArcSin[Sqrt[-(a*x)]] 
 + 1040*a^4*x^4*(-1 + a*x)^3*Sqrt[1 + a*x]*Sqrt[-(a*x*(1 + a*x))]*Hypergeo 
metric2F1[3/2, 9/2, 11/2, -(a*x)]))/((-(a*x))^(7/2)*Sqrt[1 + a*x]*Sqrt[1 - 
 a^2*x^2])
 

Rubi [A] (verified)

Time = 0.79 (sec) , antiderivative size = 164, normalized size of antiderivative = 0.62, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.458, Rules used = {6684, 6679, 109, 27, 167, 27, 167, 27, 160, 63, 222}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int e^{-3 \text {arctanh}(a x)} \left (c-\frac {c}{a x}\right )^{7/2} \, dx\)

\(\Big \downarrow \) 6684

\(\displaystyle \frac {x^{7/2} \left (c-\frac {c}{a x}\right )^{7/2} \int \frac {e^{-3 \text {arctanh}(a x)} (1-a x)^{7/2}}{x^{7/2}}dx}{(1-a x)^{7/2}}\)

\(\Big \downarrow \) 6679

\(\displaystyle \frac {x^{7/2} \left (c-\frac {c}{a x}\right )^{7/2} \int \frac {(1-a x)^5}{x^{7/2} (a x+1)^{3/2}}dx}{(1-a x)^{7/2}}\)

\(\Big \downarrow \) 109

\(\displaystyle \frac {x^{7/2} \left (c-\frac {c}{a x}\right )^{7/2} \left (-\frac {2}{5} \int \frac {a (19-3 a x) (1-a x)^3}{2 x^{5/2} (a x+1)^{3/2}}dx-\frac {2 (1-a x)^4}{5 x^{5/2} \sqrt {a x+1}}\right )}{(1-a x)^{7/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {x^{7/2} \left (c-\frac {c}{a x}\right )^{7/2} \left (-\frac {1}{5} a \int \frac {(19-3 a x) (1-a x)^3}{x^{5/2} (a x+1)^{3/2}}dx-\frac {2 (1-a x)^4}{5 x^{5/2} \sqrt {a x+1}}\right )}{(1-a x)^{7/2}}\)

\(\Big \downarrow \) 167

\(\displaystyle \frac {x^{7/2} \left (c-\frac {c}{a x}\right )^{7/2} \left (-\frac {1}{5} a \left (\frac {2}{3} \int -\frac {a (1-a x)^2 (29 a x+199)}{2 x^{3/2} (a x+1)^{3/2}}dx-\frac {38 (1-a x)^3}{3 x^{3/2} \sqrt {a x+1}}\right )-\frac {2 (1-a x)^4}{5 x^{5/2} \sqrt {a x+1}}\right )}{(1-a x)^{7/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {x^{7/2} \left (c-\frac {c}{a x}\right )^{7/2} \left (-\frac {1}{5} a \left (-\frac {1}{3} a \int \frac {(1-a x)^2 (29 a x+199)}{x^{3/2} (a x+1)^{3/2}}dx-\frac {38 (1-a x)^3}{3 x^{3/2} \sqrt {a x+1}}\right )-\frac {2 (1-a x)^4}{5 x^{5/2} \sqrt {a x+1}}\right )}{(1-a x)^{7/2}}\)

\(\Big \downarrow \) 167

\(\displaystyle \frac {x^{7/2} \left (c-\frac {c}{a x}\right )^{7/2} \left (-\frac {1}{5} a \left (-\frac {1}{3} a \left (2 \int -\frac {a (1-a x) (427 a x+1165)}{2 \sqrt {x} (a x+1)^{3/2}}dx-\frac {398 (1-a x)^2}{\sqrt {x} \sqrt {a x+1}}\right )-\frac {38 (1-a x)^3}{3 x^{3/2} \sqrt {a x+1}}\right )-\frac {2 (1-a x)^4}{5 x^{5/2} \sqrt {a x+1}}\right )}{(1-a x)^{7/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {x^{7/2} \left (c-\frac {c}{a x}\right )^{7/2} \left (-\frac {1}{5} a \left (-\frac {1}{3} a \left (-a \int \frac {(1-a x) (427 a x+1165)}{\sqrt {x} (a x+1)^{3/2}}dx-\frac {398 (1-a x)^2}{\sqrt {x} \sqrt {a x+1}}\right )-\frac {38 (1-a x)^3}{3 x^{3/2} \sqrt {a x+1}}\right )-\frac {2 (1-a x)^4}{5 x^{5/2} \sqrt {a x+1}}\right )}{(1-a x)^{7/2}}\)

\(\Big \downarrow \) 160

\(\displaystyle \frac {x^{7/2} \left (c-\frac {c}{a x}\right )^{7/2} \left (-\frac {1}{5} a \left (-\frac {1}{3} a \left (-a \left (\frac {\sqrt {x} (2525-427 a x)}{\sqrt {a x+1}}-\frac {195}{2} \int \frac {1}{\sqrt {x} \sqrt {a x+1}}dx\right )-\frac {398 (1-a x)^2}{\sqrt {x} \sqrt {a x+1}}\right )-\frac {38 (1-a x)^3}{3 x^{3/2} \sqrt {a x+1}}\right )-\frac {2 (1-a x)^4}{5 x^{5/2} \sqrt {a x+1}}\right )}{(1-a x)^{7/2}}\)

\(\Big \downarrow \) 63

\(\displaystyle \frac {x^{7/2} \left (c-\frac {c}{a x}\right )^{7/2} \left (-\frac {1}{5} a \left (-\frac {1}{3} a \left (-a \left (\frac {\sqrt {x} (2525-427 a x)}{\sqrt {a x+1}}-195 \int \frac {1}{\sqrt {a x+1}}d\sqrt {x}\right )-\frac {398 (1-a x)^2}{\sqrt {x} \sqrt {a x+1}}\right )-\frac {38 (1-a x)^3}{3 x^{3/2} \sqrt {a x+1}}\right )-\frac {2 (1-a x)^4}{5 x^{5/2} \sqrt {a x+1}}\right )}{(1-a x)^{7/2}}\)

\(\Big \downarrow \) 222

\(\displaystyle \frac {x^{7/2} \left (-\frac {1}{5} a \left (-\frac {1}{3} a \left (-a \left (\frac {\sqrt {x} (2525-427 a x)}{\sqrt {a x+1}}-\frac {195 \text {arcsinh}\left (\sqrt {a} \sqrt {x}\right )}{\sqrt {a}}\right )-\frac {398 (1-a x)^2}{\sqrt {x} \sqrt {a x+1}}\right )-\frac {38 (1-a x)^3}{3 x^{3/2} \sqrt {a x+1}}\right )-\frac {2 (1-a x)^4}{5 x^{5/2} \sqrt {a x+1}}\right ) \left (c-\frac {c}{a x}\right )^{7/2}}{(1-a x)^{7/2}}\)

Input:

Int[(c - c/(a*x))^(7/2)/E^(3*ArcTanh[a*x]),x]
 

Output:

((c - c/(a*x))^(7/2)*x^(7/2)*((-2*(1 - a*x)^4)/(5*x^(5/2)*Sqrt[1 + a*x]) - 
 (a*((-38*(1 - a*x)^3)/(3*x^(3/2)*Sqrt[1 + a*x]) - (a*((-398*(1 - a*x)^2)/ 
(Sqrt[x]*Sqrt[1 + a*x]) - a*((Sqrt[x]*(2525 - 427*a*x))/Sqrt[1 + a*x] - (1 
95*ArcSinh[Sqrt[a]*Sqrt[x]])/Sqrt[a])))/3))/5))/(1 - a*x)^(7/2)
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 63

Int[1/(Sqrt[(b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Simp[2/b   S 
ubst[Int[1/Sqrt[c + d*(x^2/b)], x], x, Sqrt[b*x]], x] /; FreeQ[{b, c, d}, x 
] && GtQ[c, 0]
 

rule 109

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_), x_] :> Simp[(b*c - a*d)*(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*((e + f 
*x)^(p + 1)/(b*(b*e - a*f)*(m + 1))), x] + Simp[1/(b*(b*e - a*f)*(m + 1)) 
 Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 2)*(e + f*x)^p*Simp[a*d*(d*e*(n - 1) 
+ c*f*(p + 1)) + b*c*(d*e*(m - n + 2) - c*f*(m + p + 2)) + d*(a*d*f*(n + p) 
 + b*(d*e*(m + 1) - c*f*(m + n + p + 1)))*x, x], x], x] /; FreeQ[{a, b, c, 
d, e, f, p}, x] && LtQ[m, -1] && GtQ[n, 1] && (IntegersQ[2*m, 2*n, 2*p] || 
IntegersQ[m, n + p] || IntegersQ[p, m + n])
 

rule 160

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_) + (f_.)*(x_) 
)*((g_.) + (h_.)*(x_)), x_] :> Simp[(b^2*d*e*g - a^2*d*f*h*m - a*b*(d*(f*g 
+ e*h) - c*f*h*(m + 1)) + b*f*h*(b*c - a*d)*(m + 1)*x)*(a + b*x)^(m + 1)*(( 
c + d*x)^(n + 1)/(b^2*d*(b*c - a*d)*(m + 1))), x] + Simp[(a*d*f*h*m + b*(d* 
(f*g + e*h) - c*f*h*(m + 2)))/(b^2*d)   Int[(a + b*x)^(m + 1)*(c + d*x)^n, 
x], x] /; FreeQ[{a, b, c, d, e, f, g, h, m, n}, x] && EqQ[m + n + 2, 0] && 
NeQ[m, -1] && (SumSimplerQ[m, 1] ||  !SumSimplerQ[n, 1])
 

rule 167

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_)*((g_.) + (h_.)*(x_)), x_] :> Simp[(b*g - a*h)*(a + b*x)^(m + 1)*(c + 
 d*x)^n*((e + f*x)^(p + 1)/(b*(b*e - a*f)*(m + 1))), x] - Simp[1/(b*(b*e - 
a*f)*(m + 1))   Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f*x)^p*Simp[b* 
c*(f*g - e*h)*(m + 1) + (b*g - a*h)*(d*e*n + c*f*(p + 1)) + d*(b*(f*g - e*h 
)*(m + 1) + f*(b*g - a*h)*(n + p + 1))*x, x], x], x] /; FreeQ[{a, b, c, d, 
e, f, g, h, p}, x] && LtQ[m, -1] && GtQ[n, 0] && IntegersQ[2*m, 2*n, 2*p]
 

rule 222

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSinh[Rt[b, 2]*(x/Sqrt 
[a])]/Rt[b, 2], x] /; FreeQ[{a, b}, x] && GtQ[a, 0] && PosQ[b]
 

rule 6679

Int[E^(ArcTanh[(a_.)*(x_)]*(n_.))*(u_.)*((c_) + (d_.)*(x_))^(p_.), x_Symbol 
] :> Simp[c^p   Int[u*(1 + d*(x/c))^p*((1 + a*x)^(n/2)/(1 - a*x)^(n/2)), x] 
, x] /; FreeQ[{a, c, d, n, p}, x] && EqQ[a^2*c^2 - d^2, 0] && (IntegerQ[p] 
|| GtQ[c, 0])
 

rule 6684

Int[E^(ArcTanh[(a_.)*(x_)]*(n_.))*(u_.)*((c_) + (d_.)/(x_))^(p_), x_Symbol] 
 :> Simp[x^p*((c + d/x)^p/(1 + c*(x/d))^p)   Int[u*(1 + c*(x/d))^p*(E^(n*Ar 
cTanh[a*x])/x^p), x], x] /; FreeQ[{a, c, d, n, p}, x] && EqQ[c^2 - a^2*d^2, 
 0] &&  !IntegerQ[p]
 

Maple [A] (verified)

Time = 0.14 (sec) , antiderivative size = 209, normalized size of antiderivative = 0.79

method	result	size
default	\(-\frac {\sqrt {\frac {c \left (a x -1\right )}{a x}}\, c^{3} \left (30 a^{\frac {9}{2}} \sqrt {-x \left (a x +1\right )}\, x^{4}+195 \arctan \left (\frac {2 a x +1}{2 \sqrt {a}\, \sqrt {-x \left (a x +1\right )}}\right ) a^{4} x^{4}+3182 a^{\frac {7}{2}} x^{3} \sqrt {-x \left (a x +1\right )}+195 \arctan \left (\frac {2 a x +1}{2 \sqrt {a}\, \sqrt {-x \left (a x +1\right )}}\right ) a^{3} x^{3}+1096 a^{\frac {5}{2}} x^{2} \sqrt {-x \left (a x +1\right )}-124 a^{\frac {3}{2}} x \sqrt {-x \left (a x +1\right )}+12 \sqrt {a}\, \sqrt {-x \left (a x +1\right )}\right ) \sqrt {-a^{2} x^{2}+1}}{30 x^{2} a^{\frac {7}{2}} \left (a x +1\right ) \sqrt {-x \left (a x +1\right )}\, \left (a x -1\right )}\)	\(209\)
risch	\(\frac {\left (15 a^{4} x^{4}+631 a^{3} x^{3}+548 a^{2} x^{2}-62 a x +6\right ) c^{3} \sqrt {\frac {c \left (a x -1\right )}{a x}}\, \sqrt {\frac {c a x \left (-a^{2} x^{2}+1\right )}{a x -1}}}{15 x^{2} \sqrt {-\left (a x +1\right ) a c x}\, \sqrt {-a^{2} x^{2}+1}\, a^{3}}+\frac {\left (-\frac {13 a^{3} \arctan \left (\frac {\sqrt {a^{2} c}\, \left (x +\frac {1}{2 a}\right )}{\sqrt {-a^{2} c \,x^{2}-a c x}}\right )}{2 \sqrt {a^{2} c}}-\frac {64 a \sqrt {-\left (x +\frac {1}{a}\right )^{2} a^{2} c +\left (x +\frac {1}{a}\right ) a c}}{c \left (x +\frac {1}{a}\right )}\right ) c^{3} \sqrt {\frac {c \left (a x -1\right )}{a x}}\, \sqrt {\frac {c a x \left (-a^{2} x^{2}+1\right )}{a x -1}}}{\sqrt {-a^{2} x^{2}+1}\, a^{3}}\)	\(245\)

Input:

int((c-c/a/x)^(7/2)/(a*x+1)^3*(-a^2*x^2+1)^(3/2),x,method=_RETURNVERBOSE)
 

Output:

-1/30*(c*(a*x-1)/a/x)^(1/2)/x^2*c^3/a^(7/2)*(30*a^(9/2)*(-x*(a*x+1))^(1/2) 
*x^4+195*arctan(1/2/a^(1/2)*(2*a*x+1)/(-x*(a*x+1))^(1/2))*a^4*x^4+3182*a^( 
7/2)*x^3*(-x*(a*x+1))^(1/2)+195*arctan(1/2/a^(1/2)*(2*a*x+1)/(-x*(a*x+1))^ 
(1/2))*a^3*x^3+1096*a^(5/2)*x^2*(-x*(a*x+1))^(1/2)-124*a^(3/2)*x*(-x*(a*x+ 
1))^(1/2)+12*a^(1/2)*(-x*(a*x+1))^(1/2))*(-a^2*x^2+1)^(1/2)/(a*x+1)/(-x*(a 
*x+1))^(1/2)/(a*x-1)
 

Fricas [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 386, normalized size of antiderivative = 1.47 \[ \int e^{-3 \text {arctanh}(a x)} \left (c-\frac {c}{a x}\right )^{7/2} \, dx=\left [\frac {195 \, {\left (a^{4} c^{3} x^{4} - a^{2} c^{3} x^{2}\right )} \sqrt {-c} \log \left (-\frac {8 \, a^{3} c x^{3} - 7 \, a c x - 4 \, {\left (2 \, a^{2} x^{2} + a x\right )} \sqrt {-a^{2} x^{2} + 1} \sqrt {-c} \sqrt {\frac {a c x - c}{a x}} - c}{a x - 1}\right ) - 4 \, {\left (15 \, a^{4} c^{3} x^{4} + 1591 \, a^{3} c^{3} x^{3} + 548 \, a^{2} c^{3} x^{2} - 62 \, a c^{3} x + 6 \, c^{3}\right )} \sqrt {-a^{2} x^{2} + 1} \sqrt {\frac {a c x - c}{a x}}}{60 \, {\left (a^{5} x^{4} - a^{3} x^{2}\right )}}, \frac {195 \, {\left (a^{4} c^{3} x^{4} - a^{2} c^{3} x^{2}\right )} \sqrt {c} \arctan \left (\frac {2 \, \sqrt {-a^{2} x^{2} + 1} a \sqrt {c} x \sqrt {\frac {a c x - c}{a x}}}{2 \, a^{2} c x^{2} - a c x - c}\right ) - 2 \, {\left (15 \, a^{4} c^{3} x^{4} + 1591 \, a^{3} c^{3} x^{3} + 548 \, a^{2} c^{3} x^{2} - 62 \, a c^{3} x + 6 \, c^{3}\right )} \sqrt {-a^{2} x^{2} + 1} \sqrt {\frac {a c x - c}{a x}}}{30 \, {\left (a^{5} x^{4} - a^{3} x^{2}\right )}}\right ] \] Input:

integrate((c-c/a/x)^(7/2)/(a*x+1)^3*(-a^2*x^2+1)^(3/2),x, algorithm="frica 
s")
 

Output:

[1/60*(195*(a^4*c^3*x^4 - a^2*c^3*x^2)*sqrt(-c)*log(-(8*a^3*c*x^3 - 7*a*c* 
x - 4*(2*a^2*x^2 + a*x)*sqrt(-a^2*x^2 + 1)*sqrt(-c)*sqrt((a*c*x - c)/(a*x) 
) - c)/(a*x - 1)) - 4*(15*a^4*c^3*x^4 + 1591*a^3*c^3*x^3 + 548*a^2*c^3*x^2 
 - 62*a*c^3*x + 6*c^3)*sqrt(-a^2*x^2 + 1)*sqrt((a*c*x - c)/(a*x)))/(a^5*x^ 
4 - a^3*x^2), 1/30*(195*(a^4*c^3*x^4 - a^2*c^3*x^2)*sqrt(c)*arctan(2*sqrt( 
-a^2*x^2 + 1)*a*sqrt(c)*x*sqrt((a*c*x - c)/(a*x))/(2*a^2*c*x^2 - a*c*x - c 
)) - 2*(15*a^4*c^3*x^4 + 1591*a^3*c^3*x^3 + 548*a^2*c^3*x^2 - 62*a*c^3*x + 
 6*c^3)*sqrt(-a^2*x^2 + 1)*sqrt((a*c*x - c)/(a*x)))/(a^5*x^4 - a^3*x^2)]
 

Sympy [F(-1)]

Timed out. \[ \int e^{-3 \text {arctanh}(a x)} \left (c-\frac {c}{a x}\right )^{7/2} \, dx=\text {Timed out} \] Input:

integrate((c-c/a/x)**(7/2)/(a*x+1)**3*(-a**2*x**2+1)**(3/2),x)
 

Output:

Timed out
 

Maxima [F]

\[ \int e^{-3 \text {arctanh}(a x)} \left (c-\frac {c}{a x}\right )^{7/2} \, dx=\int { \frac {{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} {\left (c - \frac {c}{a x}\right )}^{\frac {7}{2}}}{{\left (a x + 1\right )}^{3}} \,d x } \] Input:

integrate((c-c/a/x)^(7/2)/(a*x+1)^3*(-a^2*x^2+1)^(3/2),x, algorithm="maxim 
a")
 

Output:

integrate((-a^2*x^2 + 1)^(3/2)*(c - c/(a*x))^(7/2)/(a*x + 1)^3, x)
 

Giac [F(-2)]

Exception generated. \[ \int e^{-3 \text {arctanh}(a x)} \left (c-\frac {c}{a x}\right )^{7/2} \, dx=\text {Exception raised: TypeError} \] Input:

integrate((c-c/a/x)^(7/2)/(a*x+1)^3*(-a^2*x^2+1)^(3/2),x, algorithm="giac" 
)
 

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:sym2poly/r2sym(const gen & e,const 
index_m & i,const vecteur & l) Error: Bad Argument Value
 

Mupad [F(-1)]

Timed out. \[ \int e^{-3 \text {arctanh}(a x)} \left (c-\frac {c}{a x}\right )^{7/2} \, dx=\int \frac {{\left (c-\frac {c}{a\,x}\right )}^{7/2}\,{\left (1-a^2\,x^2\right )}^{3/2}}{{\left (a\,x+1\right )}^3} \,d x \] Input:

int(((c - c/(a*x))^(7/2)*(1 - a^2*x^2)^(3/2))/(a*x + 1)^3,x)
                                                                                    
                                                                                    
 

Output:

int(((c - c/(a*x))^(7/2)*(1 - a^2*x^2)^(3/2))/(a*x + 1)^3, x)
 

Reduce [B] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 117, normalized size of antiderivative = 0.44 \[ \int e^{-3 \text {arctanh}(a x)} \left (c-\frac {c}{a x}\right )^{7/2} \, dx=\frac {\sqrt {c}\, c^{3} i \left (780 \sqrt {a x +1}\, \mathrm {log}\left (\sqrt {a x +1}\, i +\sqrt {x}\, \sqrt {a}\, i \right ) a^{3} x^{3}+7249 \sqrt {a x +1}\, a^{3} x^{3}-60 \sqrt {x}\, \sqrt {a}\, a^{4} x^{4}-6364 \sqrt {x}\, \sqrt {a}\, a^{3} x^{3}-2192 \sqrt {x}\, \sqrt {a}\, a^{2} x^{2}+248 \sqrt {x}\, \sqrt {a}\, a x -24 \sqrt {x}\, \sqrt {a}\right )}{60 \sqrt {a x +1}\, a^{4} x^{3}} \] Input:

int((c-c/a/x)^(7/2)/(a*x+1)^3*(-a^2*x^2+1)^(3/2),x)
 

Output:

(sqrt(c)*c**3*i*(780*sqrt(a*x + 1)*log(sqrt(a*x + 1)*i + sqrt(x)*sqrt(a)*i 
)*a**3*x**3 + 7249*sqrt(a*x + 1)*a**3*x**3 - 60*sqrt(x)*sqrt(a)*a**4*x**4 
- 6364*sqrt(x)*sqrt(a)*a**3*x**3 - 2192*sqrt(x)*sqrt(a)*a**2*x**2 + 248*sq 
rt(x)*sqrt(a)*a*x - 24*sqrt(x)*sqrt(a)))/(60*sqrt(a*x + 1)*a**4*x**3)